(20) -Frenkel 87 Onsagers Spherocylinders

7/27/2019 (20) -Frenkel 87 Onsagers Spherocylinders

1/5

49122.5 can be tolerated in a fcc lattice without either both componentsbecoming localized or both fluid. We do not, however, knowanything of the crystal structures formed in these runs, nor howthey packed into the cells available to them. A full study of the

J . Phys . Chem. 1987, 91, 4912-4916problem of the crystallization of mixtures of hard spheres has yetto be made.

Acknowledgment. We thank Professor B. J . Alder for usefuldiscussions. G.J. thanks the SERC for the award of a CASE(11) Ermak, D. L.; Alder, B. J.; Pratt, L. R. J . Phys. Chem. 1981, 85 , studentship, and F.V.S. thanks the Ramsay Trustees for a Fel-3221. lowship.

Onsagers Spherocylinders RevisitedDaan FrenkelFOM Institute fo r Atomic and Molecular Physics, 1009 DB Amsterdam, The Netherlands(Received: December 4, 1986; In Final Form: February 9, 1987)

Since the work of Onsager, system s of hard spherocylinders have played a special role as the theoreticians ideal nematicliquid crystals. It is, however, well-known tha t the Ons ager theory is only a good description for extremely nonsphericalparticles. A quantitative measure for the range of validity of this theory is obtained from direct numerical calculation ofthe third through fifth virial coefficients of hard spherocylinders with length-to-width ratios, LI D, between 1 and lo6 . Th ephase diagram of spherocylinderswith realistic LID ratios can only be obtained by com puter simulation. We report moleculardynam ics and Monte Carlo simulations for spherocylinders with L ID =5 . These calculations suggest that the oldest modelfor a hard-core nematic may, in fact, have a smectic liquid-crygalline phase as well.

IntroductionLiquid crystals are probably good candidates for the titlenonsimple liquids. They ar e complex both in the microscopicna ture of their building blocks an d in th e diversity of partiallyordered phases that ma y be found in nature. Th e implicit hopeof theoreticians who propose models for liquid-crystalline meso-phases is that only one or two aspects of the microscopic structureof mesogens (liquid-crystal-forming molecules) ar e essential forthe understanding of the liquid-crystalline phases. Unfortunately,

it appears that no single structura l property c an acc ount for thewide diversity of known mesophases. W ha t is worse, there is noteven consensus about t he choice of this essential feature for anygiven phase.Onsager was the first to propose that orientational ordering ina m olecular fluid may be explained as an excluded-volume effect.The model considered by Onsager was an assembly of thinspherocylinders with length L a nd diame ter D. Onsage r showedthat , in the l imit LID - , this model system undergoes atransition from the isotropic fluid phase to an orientationallyordered (nem atic) liquid-crystalline phase. Th e physical reasonfor this phase transition is that, by form ing a nem atic, the systemcan gain translational entropy a t the expense of some orientationalentropy. Th e transition takes place at a density p* of order1/ (L2/D ) (p*B2 =3.29,2 where the second virial coefficient B,=rL2D/4). At the transition, the volume fraction occupied bythe spherocylinders is still vanishingly small (o f order D / t ) .Subsequen tly, alternative physical mechanisms have been in-voked to describe the isotropic-nematic transition. For instance,the mean-field theory of M aier a nd Sau pe3 attributes the orien-tational ordering to anisotropic intermolecular dispersion forces.Amon g experimentalists, the Maier-Sa upe theory enjoys muchgreater popularity than the Onsager description. The reason issimple: the Onsager theory yields rather poor predictions for theproperties of real thermotropic ne matics (such as the value of th enematic order parameter S and the density change A p at the(1) Onsager, L. Proc. N.Y. Acad. Sci . 1949, 51 , 627.(2) Lasher, G. J. Chem. Phys. 1970, 53 , 4141. Kayser, R. F.; Ravechi,(3) Maier, W.; Saupe, A. Z . Naturforsch. ,A 1958, 13 , 564.H. . Phys. Rev. A 1978, 17 , 2067.

isotropic-nematic tr a n ~ it io n .~ till, Onsagers model plays aunique role in the theory of liquid crystals, because it is the onlyexactly solvable model with full translational and orientationaldegrees of freedom which exhibits a transition to the nematicphase.Virial Coefficients

Unfortunately, the Onsager theory is only valid in the limit LID- , while most thermotropic liquid crystal have effective LIDratios of 3-5.5 Th e reason why the Onsager theory cannot beused to describe molecular systems with such small L ID valuesis the following: an essential assumption in its derivation is tha t,in the expansion of the free energy in powers of the density, allvirial coefficients B, with n >2 may be neglected. This conditionis satisfied if the reduced nth virial coefficient, defined asB,(reduced) Bn/B2(- I ) (1)

satisfies the condition B,(reduced) > 1.0022-3654/87/2091-4912SOl S O / O 0 1987 American Chemical Society


2/5

Onsager's Spherocylinders Revisited

loao-'rTL

io-' loa Id lo' la' lo' l o" lo"L/D + 1Figure 1. Reduced third (circles), fourth (pluses), and fifth (triangle s)virial coefficients of p rolate ha rd sphe rocy linders, as a func tion of th elength-to-breadth atio, L I D . The reduced nth virial coefficient is definedas B n/ B 2 ".In the case of th e fourth virial coefficient, B 4 / B 2 3 s shown(see Table I) .TABLE I: Reduced Third, Fourth, and Fifth Virial Coefficients ofHard Spherocylinders, as a Function of the Length-to-Breadth RatioL lD1

L I D B J B , ~ BJB? BJB?_ _ _ ~0 0.625 0.25 0.0859410 3.383 (5) X lo-' 0.5 (6) X lo-' 1.8 (1) X lo-*100 8.95 (3) X IOd2 -3.01 (3) X 2.98 ( 7 ) X1000 1.52 ( 1 ) X -5.2 (1) X 5.4 (3) X lo-'10000 2.15 (5) X IO-, -6.0 (4) X 6.0 (9 ) X lo 41000000 2.7 ( 6 ) X -0.9 (5) X lo-' 0.1 (1) X lo 4'The estimated error in th e last digit is indicated in parentheses.

extremely nonspherical shapes th at concern us here, the M onteCar lo sampling requires special care. Th e reason is the following:when avera ging over all possible configuration s of n particles, theconfigurations that contrib ute to th e nth virial coefficient are thosefor which all n particles belong to a single, irreducible cluster.Clearly, if the sampling of all particle coo rdinates is carried o utindependently, then th e probability that the resulting configurationcontribu tes to the virial coefficient becomes very small for larg eL I D. An alternative would be a sampling scheme in which then particles are initially prepared in a configuration such thatmolecules i an d i +1 overlap. Subseq uent configurations couldthen be generated by a Monte Carlo sampling of the particlecoordinates, subject to the condition that the chain is never broken.This method suffers fro m the draw back th at consecutive config-urations ar e strongly correlated. A procedure that is efficient,even for very elongated molecules, is the following. To create aconfiguration, we first fi x the orientation of molecule I . Next ,we pick a rand om orientation for molecule 2. We then constructthe excluded volume of molecules 1 and 2. (For spherocylindersthis is straightforwa rd.) Th e center of mass of molecule 2 is thenplaced at a random position within this excluded volume. Sim-ilarly, molecule 3 is made to overlap with molecule 2, and so on.Thus, we ensure that every trial configuration is statisticallyindependent. Of course, there is no guarantee that molecule nwill overlap with molecule 1. In fa ct, this probability will becomequite small for large L / D if the corresp onding virial coefficientis small. But this cannot be avoided because it is a direct con-sequence of the effect that we are looking for.Figure 1 shows the L I D dependence of B 3 / B 2 2 , 4 / B 2' ,an dB5/B$ for hard spherocylinders with LID between 1 and lo5. Thenumb ers, together with the error estimates, have been collectedin Table I (including the results for L I D = IO6, for which thestatistics a re very poor). As can be seen from Figure 1, th ecomputed virial coefficients do indeed becom e small for large LID.However, for B4 an d B, this decrease only sets in at rath er larg e

IOOOOO 2.8 (2 ) x 10-4 -0.5 (1 ) x 10-4 0.4 (2 ) x 10-4

(7) Ree, F. H. ; Hoover, W . G . J . Chem. Phys. 1964, 40 , 939.

Th eJournal of Physical Chemistry, Vol. 91 , No. 19, 1987 4913

-10 ioo id id id io' lo5 io'L / D + 1Figure 2. Asym ptotic behavior of the reduced virial coefficients of hardspherocylinders. To emphasize the different asymptotic shape depen-dence of B 3 ,on the one hand, and B4 an d B 5 ,on th e other, B , has beenmultiplied by ( L I D +2)/(log ( L I D +2)). B4 an d B5 are both multipliedby ( L I D +2 ) . Symbols: B3 ,open circles; B 4, open triangles; B 5 ,closedcircles. Estimated error bars are indicated whenever they are larger thanthe symbols.L ID values. This effect can be visualized by dividing the reducedvirial coefficients by a factor proportional to the value that oneshould expect if the asy mptotic L ID dependen ce was valid for allL l D :

B3(scaled) =( B 3 / B 2 ' ) / K D / L ) log ( L / D ) l (2a)B4(scaled) = ( B 4 / B 2 3 ) / ( D / L ) (2b)B,(scaled) =( B 5 / B $ ) / ( D / L ) (2c)

Th e resulting "scaled" virial coefficients are shown in Figure 2.Apparently, for B3 the asymptotic behavior already sets in for smallL I D . N o t so fo r B4 and B5. s can be seen from Figure 2, theapproach to the asym ptotic behavior is only observed beyond LID=O(102). The reason is that, whereas only one (positive) clusterintegral contributes to B3 ,B4 an d B5 contain several contributionsfrom diagram s of different sign and different asymptotic L I Ddepend ence. Th e partial cancellatio n of these contributions isresponsible for the behavior of B, and B5 for small L I D. In thelimit L ID- , nly the leading diag ram (i.e., the one involvingthe product fizf2zf34-.f,l) urvives. This diag ram is positive forod d n and negative for even n . Incidentally, it is worth notingthat almost identical behavior is observed for the virial coefficientsof hard ellipsoids.*The foregoing discussion confirms the generally accepted notionthat systems of hard spherocylinders with shapes resembling realliquid-crystal-forming molecules canno t be described by theOnsager theoryS9 The fact th at real nematics do not obey thepredictions of this theory is not surprising. However, this doesnot imply that excluded-volume effects cannot be responsible forphase transitions involving liquid crystals. It just m eans tha t wehave run out of exactly solvable models. For this reason, a numberof approximate theories have been proposed to describe the phasebehavior of nonspherical hard-core molecules with finite length-to-breath ratios.I0 But the problem with such approx imatetheories is tha t one cannot tell in advance whether the answersthey give are reliable. This is where computer simulation comesinto the picture.Computer Simulations

No t surprisingly, hard spherocylinders were among the firstnonspherical hard-co re systems to be studied by computer sim-(8) Frenkel, D. M o l . Phy s . 1987, 60 , 1.(9 ) Straley, J . P. Mol. Cryst. Liq. Cryst 1973, 24 , 7(IO) See. Cotter, M . A. In The Molecular Physics of Liquid Crystals;Luckhurst, G . R ,Gray, G . W. , Eds.; Academic: London, 1979 an d referencestherein. More recent references ca n be found in: Mulder, B. M ., Frenkel,D. Mol. Phys 1985, 55 , 1171.


3/5

4914 The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 FrenkelSOLID

. , , - 1.. .: .( ., .. .I . . , ,. . . .. .. , ., 7I . . ,. . I . .

< .I ., ,

I I I I I I I I II I I I I I I I II I I I I I I I II I I I I I l l lI I I I I I I I II I I I I I I I II I I I I I I t I

~ I I I I I I I I IFigure 3. Snapshots of typical molecular configurations of a system of270 parallel hard spherocylinders with LID =5 : left, nematic phase( p / p o =0.24, where po is the density of closest packing); middle, smecticphase ( p / p o=0.62);right, solid phase ( p / p o =0.89). The upper figures(A) show a projection in th e plane perpendicular to th e molecular axes,and the lower figures (B ) show a projection in a plane parallel to th emolecular axes. For the sake of clarity, the spherocylinders are indicatedby a line segment of length L .da tio n. But neither in the early simulations of Vieillard-Baron"nor in those of subsequent authorsi2 could a transition to thenematic phase be observed. The reason for this disappointing resultwas that it proved very difficult to equilibrate th e spherocylinderfluid in the density regime where th e nematic p hase is expected.']Below, we discuss a possible reason for this behavior. But fir st,let us briefly review the results of computer simulations on a closelyrelated system of nonspherical hard-cor e molecules, namely, hardellipsoids of revolution. Such sphe roids ar e characterized by asingle shape parm eter, x =a / b , where 2a is the length of thesymmetry axis of the ellipsoid and 2b is the length of an axisperpendicular to the symm etry axis. The phase diagram of hardellipsoids of revolution with axial ratios between an d 3 wasstudied by Frenkel et al. using Monte Carlo sim ~1 ati on s.l~ hesecalculations showed that, for ellipsoids with x >2.5 an d x


4/5

Onsager's Spherocylinders Revisitedcan be sim ulated, because th e volume of the periodic box scalesas L3 while the molecular volume goes as LD2. Hence, theminimum number of particles in the periodic box goes asTh e simulations reported here were carried out for a system of576 spherocylinders in an almost cubic box. Initially, thespherocylinders were prepar ed in a regu lar close-packed lattice .This lattice was generated from a face-centered-cubic hard-spherecrystal, by expansion along the [11 11 direction with a fa ctor 1+L / D . This lattice was thereupon expanded to a low density(20% of close packing), but the molecular orientation was keptfixed. Th e translation ally disorder ed fluid was thereup on com-pressed to 50% of close packing, and equilibrated configurationswere generated at a num ber of intermediate densities. For analigned spherocylinder with L I D = 5 , the transition from the"nematic" low-density phase to the smectic phase takes at a re-duced density of 0.47 (see Figure 4), where the reduced densityp(reduced) is defined as

p(reduced) =p / p o ( 3 )where po is the density at regular close packing.Next, the m olecular orientations were released. This had adrastic effect: at densities below p / p o =0.45 all orientational orderdisappea red. However, for higher densities orientation al orderpersisted. This suggests that spherocylinders with L I D =5 havean isotropic-nematic transition around p / p o =0.45. A sensitivemethod to locate the transition to the n ematic phase is to studythe pretransitional dynamics of collective orientational fluctuationsin the isotropic phase. To this end, we studied the autoc orrelationfunction C2'( t ) ,of the second-rank orientational tensor Q a s ( t ) ,defined as

with(4)

In eq 5 u, s a unit vector in the direction of the symmetry axisof molecule i. In th e isotropic phase, the correlation function C;(t )decays to zero with a characteristic decay time 7;. In the nematicphase, C2'(t)goes to a finite value as t- : C2'(t- )=1SS2,where S is the nematic order parameter. The correlation timeT2 is expected to diverge as the transition to the nematic phaseis approac hed.*' Figure 5 shows the density dependence of therate of decay of collective orientational fluctuations, as measuredby the inverse of th e correlation time 72. Th e figure clearly showsthat, for spherocylinders with L ID = 5 , the isotropic-nematictransition takes place a t 45 % of the close-packed density. This,incidentally, is the first numerical demonstration of the existenceof a nematic phase in a system of spherocylinders with a "realistic"L I D ratio.Th e next question is whether this system has a transition to asmectic phase. On ce again, the most sensitive way to probe theonset of smectic ordering is to study th e dynamics of smectic orderparameter fluctuations in the nematic phase. Th e order in asmectic A liquid crystal is characterized by a one-dimensionaldensity modulation in the direction of the molecular axes. It istherefore natural to look for pretransitional fluctuations of theFourier components of the one-particle density. An efficientmethod to detect incipient smectic ordering is to look at the decayof the longitudinal component of the intermediate scatteringfunction F(k , , t ) , defined as

F(k , J ) = (P(k,,O) p( -k , , ' ) ) ( 6 )where p (k , , t ) is the instantaneous amplitude of a longitudinaldensity fluctuation with wavevector k, . (Fo r convenience, we havechosen the nematic director parallel to the z axis). F(k, , t=O) is

(21) As the isotropic-nematic transition is first-order, the correlation timer2c aturates at a finite value at the coexistence pi n t. For a recent moleculardynamics study of rotational dynamics near the isotropic-nematic transitionof hard ellipsoids of revolution , see: Allen, M. P.; Prenkel, D. Phys . Reu. Lett . ,in press.

The Journal of Physical Chemistry, Vol. 91, N o. 19, 1987 49150.5

0.41 0

0.2 1 \ 'I

I

O 0 i . 2 "0 3 ' ' . o d " $ 5 ' "t6'' i 7\P/PW

Figure 5. Density dependence of the collective orien tational relaxationra te l / r z c triangles), for hard spherocylinders with L I D = 5 . Alsoshown in the same figure is the behavior of the relaxation rate of thelongitudinal density fluctuations with a wavevector corresponding to thefirst maximum in S ( k , ) (circles). The curves through the data pointshave been draw n as guides to the eye. The figure shows tha t the iso-tropic-nematic transition takes place around p / p o = 0.45, while thenematic has a transition to a smectic A phase at p / p o =0.58. Due toslow pretransitional fluctuations , the statistical errors in the relaxationrates become la rge in the vicinity of the phase transitions.the longitudinal part of th e static structure factor, S ( k , ) , whichdeterm ines, for instance, the X-ray scattering intensity. If atransition to a smectic phase is approached from lower densities,ther e should be smectic precursor fluctuations. The se will showup as peaks in S(k , ) , for values k , equal to (multiples of) 27r/d,where d is the spacin g of the incipient sme ctic layers. If thetransition to th e smectic phase is continuous, the peaks in S ( k , )will diverge at the transition. In the vicinity of the nem atic-smecticA transition, we expect critical slowing down of the correlatio nfunction (p(k , ,O) p ( - k , , t ) ) a t k , =k,(max), where k,(max) isthe wavevector t hat correponds to the first maximum at S ( k , ) .In Figure 5 , the rate of decay of F(k,(max),t) is plotted as afunction of density. As can be seen from the figure, there is adramatic increase with density of the lifetime of smectic orderparame ter fluctuations. The figure suggests a transition from thenematic to the smectic A phase22 at a density of -55% of closepacking. As we compress the system to still higher densities, wefind that the smectic order parame ter, which is initially very small(3% a t p / p o =0.57), starts to grow rapidly. Our simulationsindicate that t he nematic to smectic transition in this system iseither continuous or weakly first order. In real liquid crystals thenematic-smectic A transition can be either first-order or con-tinuous. A continuous transition is usually observed if the nem aticorder parame ter at t he transition is large. In the present simu-lations, the nematic order param eter at the transition is quite large(S=0.92; perfect orientational order would correspond to S =1). It therefo re seems likely tha t the nematic-smectic A phasetransition for hard spherocylinders with L I D =5 is continuous.Discussion

Systems of hard spherocylinders have long been considered the"typical" model for a nematic liquid crystal. Th e computer sim-ulations presented above indicate tha t the phase diagram of thisvery simple model system is richer th an expected . Not only dow e find a nematic p h a s e , but in a d d i t i on w e find t h a t s p h e r o -cylinders with an L I D ratio of 5 also form a smectic A phase. Inother words, pure excluded-volume effects are enough to causesmectic ordering. Th at this should be so was far from obvious.In fact, the most popular models to describe the formation ofs m e c t i c ~ ~ ~ ~ ~ ~o not even consider excluded-volume effects.(22) W e checked for other forms of ordering, in particular smectic C,(23) McMillan, W . L. Phys. Reu. A 1971, 4 , 1238.smectic B, and hexatic B, but found no evidence fo r such ordering.


5/5

4916 J. Phys . Chem. 1987, 91, 4916-4922The fact that h ard spherocylinders can have both a nematicand a smectic phase may explain the problems that Vieillard-Baron experienced in his search for the isotropic-nematictransition in a system of 616 hard spherocylinders with L ID =2. Vieillard-Baron found in his Mo nte Ca rlo simulations thatextremely long preparatory runs were required to generate anequilibrated (isotropic) fluid configuration, starting from thecrystalline solid. In retrospect, it seems likely that Vieillard-Barons simulations w ere plagued by critical slowing-down effectsdue to the vicinity of both the isotropic-nematic and the ne-matic-smectic phase transitions. Wh ethe r the system studied by

Vieillard-Baron does in fac t have thermody namica lly stable ne-matic and smectic phases is still an open question. Simulatio nson spherocylinders with L I D ratios other than 5 are currentlyunder way.To conclude, let us return to Onsagers model system consistingof long, thin spherocylinders. As mentioned earlie r, this modelplays a special role in the statistical mechanical theory of liq-(24) Dowell, F.Phys. Rec. A 1983, 28, 3526.

uid-crystal formation because it is the only exactly solvable modeltha t predicts a transition to the nematic phase. However, in thestrongly aligned nem atic phase th e second virial theory is nolonger valid (see footnote 6). Actually, a breakdown of theOnsager approximation at high densities is to be expected, becauseat sufficiently high packing fraction thin spherocylinders shouldform a crystal, rath er than a liquid crystal. However, the presentsimulations on spherocylinders with a finite L I D ratio (bothaligned and nonaligned) suggest that, before they freeze, Onsagersspherocylinders may well form a smectic liquid crystal.From the work of Rahm an, I learned tha t the complex be-havior of many-body systems can often be explained in terms ofvery simple intermolecular interactions. It is gratifying to see thatthis appro ach is fruitfu l, even for systems as complex as liquidcrystals.

Acknowledgment. I gratefully acknowledge many stimulatingdiscussions with Bela Mulder, Henk Lekkerkerker, and AlainStrooba nts. Mike Allen contributed most to th e development ofa vectorizable molecular dynamics code for nonspherical hard-coremolecules which was used in this work.

Frequency-Dependent Specific Heat in a Simulation of the Glass TransitionGary S. Crest*Corporate Research Science Laboratory, Exxon Research and Engineering Company,Annandale, New Jersey 08801and Sidney R. Nagel*The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637(Received: December 4, 1986; In Final Form: February 19, 1987)

Recent experiments have shown that there is a frequency dependence to the specific heat of liquid glycerol as it is cooledthrough the glass transition. We show that in a simulation of a simple Lennard-Jones fluid the same phenomenon can beobserved. The re is a large difference between the behavior of the constant pressure and constant volume specific heats. Theformer shows a pronounced drop as the tem perature is lowered while the latter shows very little signature of the transitionat all

IntroductionTh e glass transition o ccurs when a liquid is cooled sufficientlyfar below its equilibrium freezing temperature so as to becomean amo rphou s solid. To reach this glass state the liquid must besupercooled rapidly in order to prevent the nucleation of thecrystalline phase. The na ture of the glass transition is still opento debate. For example, there is ongoing controversy over whetherall classical liquids can for m a glass if cooled rapidly enough orwheth er only liquids which have certain pro perties can be glass

formers. It is also unsettled wh ether th e glass transition is a truephase transition or whether it is simply a kinetic freezing of theliquid.Experimental attemp ts at studying this phenomenon have beenhampered by one of the essential characteristics of the glasstransition itself. Th e liquid becomes so viscous as the transitiontemperature, T grs approached tha t it becomes progressively moredifficult to keep the sample in equilibrium. Thus the time scaledetermined by th e experiment depends either on the cooling rateor on the frequency of the probe being used and static susceptibilitycannot be measured near T B . On the other hand, theoreticalinvestigations of the glass transition have not been able to agreeon the microscopic mech anism which is responsible for th e phe-nomenon. Should the same mechanism be responsible for the glassformation i n polymers as well as in metals or does one need0022-365418712091-4916$01.50/0

different theories for different types of systems?It may seem paradoxical that a computer simulation of the glasstransition could provide much help at sorting out these variousissues. After all, glass transitions ar e inextricably bound up withlong relaxation times whereas comp uter simulations can only berun for relatively short times. Th e slowest cooling rat e used ina computer simulation is much faster than ca n be achieved in eventhe most rapid experimental quench. The simulation is thus forcedto study glass formation very far away in tem perature from whatwould occur in an ideal, infinitely slow, cooling rate experiment.Another problem with simulations is that they usually are onsystems with very simple interparticle interactio ns. It is muchmore difficult to simulate a liquid of complicated molecules in-teracting via long-range forces.Despite these drawba cks compute r simulations have provideda number of insights about the glass state and about the transitionitself. Rah man and co-workers studied the properties of a frozenLennard-Jones fluid and showed that it behaved much like a solid.They also investigated the nucleation rates for various potentialsand showed that in some cases the sample could remain amorphousduring very long simulation^.^^^ I t was also found4 that the

( I ) Rahman, A .; Mandell, M . J.; McTague, J. P. J . Ch em. P h y s . 1976,64, 1564.0 987 American Chemical Society

Documents

(20) -Frenkel 87 Onsagers Spherocylinders